The angle of elevation of the top of a tree from a point on the ground 60m away from the foot of the tree is…

1. Analyze the question

We are given:

• The distance from the foot of the tree to the point on the ground (adjacent side) = $60\text{m}$
• The angle of elevation ($\theta$) = $78^\circ$
• We need to find the height of the tree (opposite side).

2. Explain the concept

This is a trigonometry problem involving a right-angled triangle. We use the SOH CAH TOA mnemonic:

• SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Since we have the Adjacent side and need the Opposite side, we use the Tangent (tan) ratio.

3. Show the solution

Step 1: Set up the formula

Step 2: Solve for $h$

Step 3: Calculate the value Using a calculator, $\tan(78^\circ) \approx 4.7046$

Step 4: Round to the nearest whole number Rounding $282.276$ to the nearest whole number gives $282\text{m}$.

4. Identify the answer

Comparing our result to the given options: A. 148m B. 382m C. 282m D. 248m

The correct option is C.

5. Exam Tip

When dealing with "angle of elevation" problems, always draw a quick sketch of a right-angled triangle. Remember that the "foot of the tree" is the base, and the "height" is the vertical side. If your calculator gives you a very strange number, ensure it is set to DEG (Degrees) mode and not RAD (Radians) mode!